User:MikeRumex/sandbox/ordered vector space

A point x in R² and the set of all y such that x≤y (in red). The order here is x≤y if and only if x₁ ≤ y₁ and x₂ ≤ y₂.

In mathematics an ordered vector space is a vector space equipped with a preorder or partial order which is compatible with the vector space operations.

Definition

Given a vector space V over the real numbers R and a preorder ≤ on the set V, the pair (V, ≤) is called an preordered vector space if for all x,y,z in V and 0 ≤ λ in R the following two axioms are satisfied

1. x ≤ y implies x + z ≤ y + z
2. y ≤ x implies λ y ≤ λ x.

If, in addition, ≤ is a partial order, then (V, ≤) is called an partially ordered vector space.

Cones and preorders

If V is a vector space over the real numbers, a subset $C\subseteq V$ is called a cone if

1. $C+C\subseteq C$
2. $\lambda C\subseteq C$ for all $\lambda \geq 0$

The set $C\subseteq V$ is called a proper cone if $C\cap C=\{0\}$.

If (V, ≤) is a preordered vector space the set $V_{+}:=\{v\in V:v\geq 0\}$ is a cone, and if ≤ is a partial order, $V_{+}$ is a proper cone. Conversely, if $C\subseteq V$ is a cone and defining $x\leq _{C}y$ to mean $y-x\in C$, then $(V,\leq _{C})$ is a preordered vector space, and a partially ordered vector space if $C$ is a proper cone. In this way there is a bijection between all cones (proper cones) and all preorders (partial orders) on V that are translation invariant and positively homogeneous.

Examples

• The real numbers with the usual order is an ordered vector space.
• R² is an ordered vector space with the ≤ relation defined in any of the following ways (in order of increasing strength, i.e., decreasing sets of pairs):
  • Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order. The positive cone is given by x > 0 or (x = 0 and y ≥ 0), i.e., in polar coordinates, the set of points with the angular coordinate satisfying -π/2 < θ ≤ π/2, together with the origin.
  • (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order of two copies of R with "≤"). This is a partial order. The positive cone is given by x ≥ 0 and y ≥ 0, i.e., in polar coordinates 0 ≤ θ ≤ π/2, together with the origin.
  • (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of two copies of R with "<"). This is also a partial order. The positive cone is given by (x > 0 and y > 0) or (x = y = 0), i.e., in polar coordinates, 0 < θ < π/2, together with the origin.
  • (a,b) ≤ (c,d) if and only if a ≤ c, is a preorder.

Only the second order is, as a subset of R⁴, closed, see partial orders in topological spaces.

For the third order the two-dimensional "intervals" p < x < q are open sets which generate the topology.

• A Riesz space is a partially ordered vector space where the order gives rise to a lattice, i.e., for every pair of elements there exists a supremum.
• The space of continuous functions on [0,1] where f ≤ g iff f(x) ≤ g(x) for all x in [0,1] is a Riesz space. When this space is also endowed with the uniform norm this space becomes a Banach lattice.
• Rⁿ becomes a partially ordered vector space when endowed with the standard order, i.e., x ≤ y if and only if x_i ≤ y_i for all i = 1, … , n.
• R³ with partial order $\leq _{C}$ defined from the Lorentz cone (aka ice-cream cone) $C:=\{x:x_{1}\geq {\sqrt {x_{2}^{2}+x_{3}^{2}}}\}$, i.e., $x\leq _{C}y$ means $y-x\in C$, is a partially ordered vector space. This is one of the simplest partially ordered vector spaces that is not a vector lattice. Suprema of arbitrary pairs of elements do not necessarily exist, e.g., the pair of vectors (0,0,0) and (0,0,1) have no supremum with respect to the partial order $\leq _{C}$.

See also

• Partially ordered space
• Ordered Banach space
• Banach lattice
• Riesz space

References

• Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
• Schaefer, Helmut H (1999). Topological vector spaces, 2nd ed. New York: Springer. pp. 204–205. ISBN 0-387-98726-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Aliprantis, Charalambos D (2003). Locally solid Riesz spaces with applications to economics (Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0-8218-3408-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Category:Functional analysis Category:Ordered groups